Divide $ \dfrac{5x^2+10x}{x^2+7x+12} $ by $ x^2-4 $ to get $ \dfrac{5x}{x^3+5x^2-2x-24} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{5x^2+10x}{x^2+7x+12} }{x^2-4} & \xlongequal{\text{Step 1}} \frac{5x^2+10x}{x^2+7x+12} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2-4}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 5x \cdot \color{blue}{ \left( x+2 \right) } }{ x^2+7x+12 } \cdot \frac{ 1 }{ \left( x-2 \right) \cdot \color{blue}{ \left( x+2 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5x }{ x^2+7x+12 } \cdot \frac{ 1 }{ x-2 } \xlongequal{\text{Step 4}} \frac{ 5x \cdot 1 }{ \left( x^2+7x+12 \right) \cdot \left( x-2 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ 5x }{ x^3-2x^2+7x^2-14x+12x-24 } = \frac{5x}{x^3+5x^2-2x-24} \end{aligned} $$

Divide $ \dfrac{5x}{x^3+5x^2-2x-24} $ by $ x^2+x-6 $ to get $ \dfrac{5x}{x^5+6x^4-3x^3-56x^2-12x+144} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{5x}{x^3+5x^2-2x-24} }{x^2+x-6} & \xlongequal{\text{Step 1}} \frac{5x}{x^3+5x^2-2x-24} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2+x-6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 5x \cdot 1 }{ \left( x^3+5x^2-2x-24 \right) \cdot \left( x^2+x-6 \right) } \xlongequal{\text{Step 3}} \frac{ 5x }{ x^5+x^4-6x^3+5x^4+5x^3-30x^2-2x^3-2x^2+12x-24x^2-24x+144 } = \\[1ex] &= \frac{5x}{x^5+6x^4-3x^3-56x^2-12x+144} \end{aligned} $$